Communications in Mathematical Physics

, Volume 246, Issue 3, pp 453–472 | Cite as

Equivalence of Additivity Questions in Quantum Information Theory

  • Peter W. Shor
Article

Abstract

We reduce the number of open additivity problems in quantum information theory by showing that four of them are equivalent. Namely, we show that the conjectures of additivity of the minimum output entropy of a quantum channel, additivity of the Holevo expression for the classical capacity of a quantum channel, additivity of the entanglement of formation, and strong superadditivity of the entanglement of formation, are either all true or all false.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Koenraad, M.R., Audenaert, Braunstein, S.L.: On strong superadditivity of the entanglement of formation. quant-ph/0303045Google Scholar
  2. 2.
    Bell, J.: On the Einstein Podolsky Rosen paradox. Physics 1, 195–200 (1964)MATHGoogle Scholar
  3. 3.
    Benatti, F., Narnhofer, H.: Additivity of the entanglement of formation. Phys. Rev. A 63, art. 042306 (2001)Google Scholar
  4. 4.
    Bennett, C.H., Bernstein, H.J., Popescu, S., Schumacher, B.: Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046–2052 (1996)Google Scholar
  5. 5.
    Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824–3851 (1996), quant-ph/9604024CrossRefMathSciNetGoogle Scholar
  6. 6.
    Gordon, J.P.: Noise at optical frequencies: Information theory. In: Proceedings of the International School of Physics Enrico Fermi. Course XXXI: Quantum Electronics and Coherent Light, P.A. Mills, (ed.), New York: Academic Press, 1964), pp. 156–181Google Scholar
  7. 7.
    Hayden, P.M., Horodecki, M., Terhal, B.M.: The asymptotic entanglement cost of preparing a quantum state. J. Phys. A: Math. Gen. 34, 6891–6898 (2001)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Holevo, A.S.: Bounds for the quantity of information transmitted by a quantum communication channel. Probl. Peredachi Inf. 9(3), 3–11 (1973) [in Russian; English translation in Probl. Inf. Transm. (USSR) 9, 177–183 (1973)]Google Scholar
  9. 9.
    Holevo, A.S.: The capacity of the quantum channel with general signal states. IEEE Trans. Info. Theory 44, 269–273 (1998)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    King, C., Ruskai, M.B.: Minimal entropy of states emerging from noisy quantum channels. IEEE Trans. Info. Theory 47, 192–209 (2001)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Levitin, L.B.: On the quantum measure of the amount of information. In: Proceedings of the Fourth All-Union Conference on Information Theory, Tashkent (1969), pp. 111–115, (in Russian)Google Scholar
  12. 12.
    Matsumoto, K., Shimono, T., Winter, A.: Remarks on additivity of the Holevo channel capacity and of the entanglement of formation. quant-ph/0206148Google Scholar
  13. 13.
    Pomeransky, A.: Strong superadditivity of the entanglement of formation follows from its additivity. quant-ph/0305056Google Scholar
  14. 14.
    Ruskai, M.B.: Some bipartite states do not arise from channels. quant-ph/0303141Google Scholar
  15. 15.
    Schumacher, B., Westmoreland.: Sending classical information via a noisy quantum channel. Phys. Rev. A 56, 131–138 (1997)CrossRefGoogle Scholar
  16. 16.
    Shor, P.W.: Capacities of quantum channels and how to find them. quant-ph/0304102Google Scholar
  17. 17.
    Vidal, G., Dür, W., Cirac, J.I.: Entanglement cost of mixed states. Phys. Rev. Lett. 89, art. 027901 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Peter W. Shor
    • 1
    • 2
  1. 1.AT & Labs ResearchUSA
  2. 2.Dept. of MathematicsMassachusetts Institute of TechnologyUSA

Personalised recommendations